morlet wavelet

How Does the Morlet Wavelet Work?

The Morlet wavelet is defined by a complex exponential modulated by a Gaussian function. Mathematically, it can be represented as:
\[ \psi(t) = \pi^{-1/4} e^{i \omega_0 t} e^{-t^2 / 2} \]
where \( \omega_0 \) is the central frequency. The wavelet transform of a time series \( x(t) \) is given by:
\[ W_x(s, \tau) = \int_{-\infty}^{\infty} x(t) \psi^*\left(\frac{t-\tau}{s}\right) dt \]
Here, \( s \) represents the scale (related to frequency), and \( \tau \) represents the time shift. The resulting coefficients \( W_x(s, \tau) \) provide a time-frequency representation of the data.

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